Sure, let's match the like items step by step.
We have two groups of terms:
- Group 1:
a. [tex]\(4xy^3\)[/tex]
b. [tex]\(14xy\)[/tex]
c. [tex]\(15y^2\)[/tex]
d. [tex]\(-6\)[/tex]
e. [tex]\(5x^2\)[/tex]
- Group 2:
1. [tex]\(-12x^2\)[/tex]
2. [tex]\(6xy\)[/tex]
3. 11
To match like items, we look for terms that have the same variable composition and similar degrees. Let's analyze each pair:
1. Matching term e from Group 1 with terms from Group 2:
- e. [tex]\(5x^2\)[/tex] has the term [tex]\(x^2\)[/tex].
- Matching with Group 2:
- 1. [tex]\(-12x^2\)[/tex] has the term [tex]\(x^2\)[/tex].
- Hence, [tex]\(5x^2\)[/tex] matches with [tex]\(-12x^2\)[/tex].
2. Matching term b from Group 1 with terms from Group 2:
- b. [tex]\(14xy\)[/tex] has the term [tex]\(xy\)[/tex].
- Matching with Group 2:
- 2. [tex]\(6xy\)[/tex] has the term [tex]\(xy\)[/tex].
- Hence, [tex]\(14xy\)[/tex] matches with [tex]\(6xy\)[/tex].
Other terms in Group 1 do not find exact matches in Group 2:
- a. [tex]\(4xy^3\)[/tex] does not match with any options in Group 2.
- c. [tex]\(15y^2\)[/tex] does not match with any options in Group 2.
- d. [tex]\(-6\)[/tex] does not match with 11 in Group 2 (as they are different numbers).
Now, we match the pairs and provide the result in a dictionary format:
- [tex]\(14xy\)[/tex] matches with [tex]\(6xy\)[/tex]: therefore, term b matches with item 2.
- [tex]\(5x^2\)[/tex] matches with [tex]\(-12x^2\)[/tex]: therefore, term e matches with item 1.
The final matches are:
```json
{
"2": "b", // [tex]\(14xy\)[/tex] matches with 2. [tex]\(6xy\)[/tex]
"1": "e" // [tex]\(5x^2\)[/tex] matches with 1. [tex]\(-12x^2\)[/tex]
}
```
Therefore, the answer indicating the matches is:
```json
{'2': 'b', '1': 'e'}
```
